Planning a trip to Stockholm: a few questions?

So my friend has an apartment in Hornstull, which is a neighborhood in the Western part of Sodermalm in Stockholm. His parents are picking us up from the airport, but they'll be staying at their country house there. We're in Stockholm for about nine days, but we were thinking about taking an overnight train to Copenhagen, Oslo, or Helsinki to stay for a bit while we're there. Which of these cities might be the best destination? I've heard that it would be Copenhagen, but I couldn't really say. We're looking to check out the city and get experience the local culture, as Stockholm is supposed to be the most museum dense city in the world. What are some interesting things to check out? I'll add that there are three of us going, and we're all 18; take that into account with respect to your suggestions, please, haha. We also want to check out some of the local nightlife, though to my understanding very little of it is open to people our age. Are there clubs and the like that would cater to us? If so, what would be the procedure for that? Finally, we have a pretty loose budget for the trip, so we can splurge a bit, but I'd rather not blow money on things that aren't worth it. Thanks for any help you can give with these questions!